
If the generic dimension of \(U\) -stabilisers on \(Z_{\min }\)  is zero, then one does not need to blow up all of \(Z_{\min }\)  and there is an explicit description of the open subset admitting a quotient given by [1]}.

If \(U\)  is abelian, but the generic dimension of the \(U\) -stabilisers on \(Z_{\min }\)  is positive, then either

the generic dimension of the \(U\) -stabilisers on \(Z_{\min }\)  equals the generic dimension of the \(U\) -stabilisers on \(Y_{\min }\)  or

the generic dimension of the \(U\) -stabilisers on \(Z_{\min }\)  is bigger than the generic dimension of the \(U\) -stabilisers on \(Y_{\min }\) .

In the first case, one does not need to blow up all of \(Z_{\min }\)  and it is easier to describe the open subset that one obtains a quotient of; see \(§\)REF  for a situation in which this is particularly straight-forward. In the second case, one must blow-up all of \(Z_{\min }\)  and use jets to describe the open subset admitting a quotient.

For general \(U\)  such that the generic dimension of the \(U\) -stabilisers on \(Z_{\min }\)  is positive, the results of the blow-up procedure is described by [1]}. In this case one fixes a series of subgroups of \(U\)  which are normal in \(H\)  and whose successive quotients are abelian with \(\lambda (\mathbb {G}_m)\)  acting by a single weight. One could then perform quotients in stages doing blow-ups at each stage to arrange for the stabilisers to be constant dimension on \(Y_{\min }\)  (cf. [1]}).

